Low Complexity Power Allocation Schemes in Regenerative Multi-user Relay Networks
11 Low Complexity Power Allocation Schemes inRegenerative Multi-user Relay Networks
Arvind Chakrapani , Robert Malaney , and Jinhong Yuan Qualcomm Flarion Technologies, Bridgewater, New Jersey, USA [email protected] School of Electrical Engineering and Telecommunication, University of New SouthWales, Sydney, Australia { r.malaney,jinhong.yaun } @unsw.edu.au Abstract
In relay assisted wireless communications, the multi-source, single relay and single destination system (an M -1-1 system) is of growing importance, due to the increased demand for higher network throughput and connectivity.Previously, power allocation in M -1-1 systems have assumed availability of instantaneous channel state information(CSI), which is rather idealistic. In this paper we consider an M -1-1 Decode-and-Forward (DF), Full-Duplex,orthogonal frequencey division multiple access (OFDMA) based relay system with statistical-CSI and analyze theachievable rate R of such a system. We show how R can only be maximized by numerical power allocationschemes which has a high-complexity of order O ( M ) . By introducing a rational approximation in the achievablerate analysis, we develop two low-complexity power allocation schemes that can obtain a system achievable ratevery close to the maximum R . Most importantly, we show that the complexity of our power allocation schemes isof order O ( M log M ) . We then show how our power allocation schemes are suitable for a multi-user relay system,where either the priority is to maximize system throughput, or where lower computations in power allocationscheme are essential. The work we present in this paper will be of value to the design and implementation ofreal-time multi-user relay systems operating under realistic channel conditions. A part of this work was submitted to the IEEE International Conference on Communications (ICC), Miami, FL, USA, Jun.10-Jun.14,2014. In this paper, we provide results with different low complexity power allocation schemes in addition to the ones proposed in [3]. a r X i v : . [ c s . I T ] A ug I. I
NTRODUCTION
Recently there has been significant interest from both academia and industry in the concept of cooper-ative relaying in infrastructure based broadband wireless access for 4G networks, e.g., 802.16j - MobileMultihop Relay (MMR) specification [1]. In relay assisted communication, relay stations (RS), either fixedor mobile, are introduced to increase the capacity (for both uplink and downlink) or connectivity amongmobile sources (MS). The 802.16j MMR standard, specifies two modes of relaying techniques. One is thetransparent relaying mode, where relays are used to increase capacity of MS who are within the range ofthe Base Station (BS). The other is the non-transparent relaying mode, where relays are deployed mainlyto increase the coverage area of the BS. In this work, we will focus on the uplink of a cellular systemwith transparent mode of relaying.In the transparent mode, the relay is used to enhance the throughput of each source. The relaying canemploy either the amplify and forward (AF) or the DF strategy. It was shown in [2] that DF providesa higher achievable rate relative to AF at low signal-to-noise ratio (SNR). Also, the relay can operatein either full-duplex (sources and relay transmit simultaneously) or half-duplex mode (sources and relaytransmit during different time slots). Note that half-duplex mode can be implemented with a single antennawhereas full-duplex mode will require additional antennas for self-intereference cancellation. It has beenpreviously shown (e.g., [5]) that the full-duplex mode of relaying is spectrally more efficient than thehalf-duplex mode. A general precipt so far has been that the practical implementation of full-duplex modeis often not possible due to large difference in transmit and receive power at the relay.However, recently there have been significant works both from academics (e.g., [6]-[12]]) and industry(e.g., [14]-[16]) regarding the feasibility of a full-duplex relay system with DF strategy. In fact, the worksin [8] and [9] showed that a practical full-duplex system can be built using off-the-shelf hardware. Antennaseperation with Analog/Digital Cancellation techniques was used in [8] and the experiments showed thatthe full-duplex mode can be practically implemented. A novel self intereference cancellation technique wasused in [9] and a working prototype was developed, which achieved median performance that was within8 % of an ideal full-duplexing system. Also, in [17] a transmission policy based on block Markov encodingfor a DF full-duplex relay system was described. The above works have clearly demonstrated that buildinga practical DF full-duplex relay system without introducing significant latency into the transmission, is indeed possible. In this paper, we consider a DF full-duplex relay system with perfect self-interferencecancellation.In the transparent mode, the single source, relay and destination form an 1-1-1 system. The 1-1-1system has been well investigated over the years (e.g. [18]-[21] and the references therein) and variousstudies have focused on several performance aspects, including achievable rates [19], outage probabilities[11][12][13] and power allocation [11][13][20]. Recently the performance of an M -1-1 system has gainedmuch attention [22][23][24]. Resource allocation and relay selection in a multi-user OFDMA based systemwas studied in [22], assuming access to full-CSI. The power allocation scheme for a multi-source AF relaysystem to maximize the network throughput was investigated in [23]. The multi-source achievable rateand power allocation for a half-duplex relay system was investigated for a multiple access relay channelin [24], assuming availability of full-CSI at the relay.However, none of the works have investigated practical power allocation schemes in an M -1-1 DF relaysystem in Rayleigh fading environment, when no instantaneous CSI is available, or where only statisticalinformation of the channel state (i.e., statistical CSI) is available. This is mainly due to the complicatednature of the throughput analysis and as such only numerical methods of optimal power allocation canbe employed, which have a complexity of order O ( M ) [30]. Such complexity renders them infeasiblefor implementation in real-time systems, especially when the number of users in the system increase.Note that in a 4G network, the number of users M in the network is typically large [1] and efficientpower allocation at the relay will be lead to significant increase in the system throughput. In this paper,we develop two low-complexity power allocation schemes (of different computational speed) at the relayfor an M -1-1 system with statistical-CSI. Let R P AS − be the maximum achievable rate of the systemobtained with an optimal power allocation scheme (we denote this as PAS-0) at the relay. We show howour power allocation schemes can obtain a system achievable rate close to R P AS − , and show that thecomplexity of our power allocation schemes is of order O ( M log M ) .Our contributions reported in this paper are as follows. First, we analyze the achievable rate R an M -1-1system with statistical-CSI. Second (and the key contribution in this paper), is that we introduce a rationalapproximation in the achievable rate analysis, which helps us develop low-complexity power allocationschemes that can obtain a system throughput close to R P AS − . Third, using our rational approximation, we develop two low-complexity power allocation schemes (of varying computational speed) at the relay.More specifically, we develop the following two power allocation schemes. • We develop a Lagrangian-based power allocation scheme (we denote this as PAS-1) that obtains anachievable rate R P AS − , which is approximately equal to R P AS − for all practical purposes. We showthrough analysis and simulations that PAS-1 has negligible loss in throughput compared to PAS-0. Mostimportantly we show that the complexity of the PAS-1 algorithm is of order O ( M log M ) . • Utilizing the results from the PAS-1 algorithm, we develop a second power allocation scheme (wedenote this as PAS-2) which delivers a system achievable rate within ≈ − of R P AS − and requireslower computations compared to PAS-1 and is free of logarithmic and cube root operations.The paper is organized as follows. In section III, we analyze the achievable rate R of an M -1-1 systemwith statistical-CSI. In section IV, we provide the approximations required for our new power allocationschemes. In section V, we develop a Lagrangian-based power allocation scheme (PAS-1) that obtains asystem achievable rate approximately equal to R P AS − for all practical purposes. In section VI we developthe second power allocation scheme (PAS-2), which provides a system achievable rate within ≈ − of R P AS − and requires lower computations compared to the PAS-1 algorithm. In section VII, we discussthe computational complexity of our two power allocation schemes. In section VIII we provide analyticaland simulations results. Finally, in section IX, we draw conclusions.II. S YSTEM M ODEL
Consider a multi-source relay system shown in Fig. 1. Sources S m , m ∈ { , ...M } transmit theirinformation to the destination d simultaneously with the help of a full-duplex relay r . A bin indexingscheme as in [19] was assumed to transmit information and parity bits. The conventional DF relayingwith orthogonal transmission through OFDMA is assumed. With OFDMA, the m th source S m transmitsits messages in the frequency bands f m , the relay r receives and transmits at frequencies f , ..., f M ,respectively. Note that one antenna is sufficient at the relay for transmitting/receiving an OFDMA signalwith M sub channels. The destination receives signals at these M orthogonal frequency bands. With theseconstraints, the multi-source system can be viewed as M independent parallel 1 - 1 - 1 triangle systems, Here, by orthogonal transmission we mean there is no interference at the destination due to transmissions from multiple sources andrelay. one of which is shown in Fig. 2. All channels are assumed to undergo Rayleigh fading and are corruptedwith Additive White Gaussian Noise (AWGN). The source S m transmits in the frequency bands f m to thedestination. The source S m begins by encoding a q -bit message Qm into a codeword of length n ( k < n ).The codeword is then divided into B blocks of length nr c bits each, where r c is channel coding rate at theencoder ( r c ≤ ). The coding rate r c specifies how much redundancy is transmitted with every messagebit. For a q -bit message, q/r bits are transmitted in B + 1 blocks. The codeword is encoded into c symbols x [1] , ..., x [ c ] and transmitted over the channel, under the power constraint (cid:12)(cid:12)(cid:12) c (cid:80) cj =1 x [ j ] (cid:12)(cid:12)(cid:12) ≤ P s , where P s is the maximum transmit power available at each source. The relay decodes and forwards a new block x [ j ] to aid the communication between source and destination. x [ j ] is also encoded into c symbolssubject to the power constraint (cid:12)(cid:12)(cid:12) c (cid:80) cj =1 x [ j ] (cid:12)(cid:12)(cid:12) ≤ P m , where P m is the power allocated by the relay fortransmitting the m th source signal. The received signal at the relay y rm and the destination y dm are givenby, y rm = C srm x [ j ] + n r (1) y dm = C sdm x [ j ] + C rdm x [ j ] + n d , (2)where C srm , C srm and C srm represent the channel gains between S m to r (denoted as S-R), S m to d (denoted asS-D), and r to d (denoted as R-D), respectively. Here, n r and n d are independent AWGN’s with zero meanand unit variance. We consider a propagation model as in [17] and let, C srm = | h srm | ( d srm ) α N r , C sdm = | h sdm | ( d sdm ) α N d and C rdm = | h rdm | ( d rdm ) α N d , where h srm , h sdm and h rdm are complex fading random variables for channels between S m to r , S m to d , and r to d , respectively. N r and N d are the noise spectral densities at the relay andat the destination respectively. Here, d sdm , d srm and d rdm represent the normalized distances between S-D,S-R and R-D respectively. Note that the distances are normalized with respect to a reference distance of d = 1 unit. Here, α represents the pathloss exponent. For a Rayleigh channel, the real and imaginaryparts of the complex fading variables are Gaussian distributed having zero mean and variance 1/2. Note that, here we have ignored the effect of shadowing on the channel gain for simplification. Including the shadowing componentwould scale down the achievable rate by a constant factor, but does not add any further insights.
A. Problem Statement
Consider an M -1-1 system described above. Let the relay have a maximum total transmit power of P r and the transmit power P s at each source be fixed. We investigate the following two problems. Whatis the achievable rate of the M -1-1 DF relay system when all channels undergo Rayleigh fading? Letthe relay allocate power among M sub-channels as { P , ..., P M } , such that (cid:80) Mm =1 P m = P r . What is thepower allocation vector { P , ..., P M } at the relay which obtains the achievable rate R ?III. A CHIEVABLE R ATE WITH S TATISTICAL -CSIThe m th source, the relay and the destination, form a 1-1-1 system as shown in Fig. 2. The instantaneousachievable rate for such a 1-1-1 system can be expressed as ([17], Equation (4.33)), R im = min (cid:40) log (cid:32) | h srm | P s ( d srm ) α N r (cid:33) , log (cid:32) (cid:12)(cid:12) h sdm (cid:12)(cid:12) P s ( d sdm ) α N d + (cid:12)(cid:12) h rdm (cid:12)(cid:12) P m ( d rdm ) α N d (cid:33)(cid:41) . (3)Note that (3) is valid (see [17], section 4.2.5) only for a fading channel where the phase is uniformlydistributed over [0 , π ) (e.g., Rayleigh Fading channel), i.e., there is no correlation between the relaysignal and the source signal. Note also that throughout this paper log( · ) represents logarithm to base2. The achievable rate of an 1-1-1 system with averaged over all channel fading states (with Rayleighdistribution), i.e., with statistical-CSI is given by [25], R m = min { R m , R m } , where, R m = log( e ) (cid:20) exp (cid:18) k srm P s (cid:19) E (cid:18) k srm P s (cid:19)(cid:21) , (4)and R m = log( e ) (cid:104) P m k sdm exp (cid:16) k rdm P m (cid:17) E (cid:16) k rdm P m (cid:17) − P s k rdm exp (cid:16) k sdm P s (cid:17) E (cid:16) k sdm P s (cid:17)(cid:105) ( P m k sdm − P s k rdm ) , (5)where, k srm = ( d srm ) α N r , k sdm = ( d sdm ) α N d and k rdm = ( d rdm ) α N d and where, E ( · ) is the exponential integraldefined as E ( x ) = (cid:82) ∞ e − xt t dt, ( x > and e = exp(1) ≈ . . It will be useful to rewrite (5) as, R m = R +2 m + R − m , where, R +2 m = log( e ) exp (cid:16) k rdm P m (cid:17) E (cid:16) k rdm P m (cid:17) − exp (cid:16) k sdm P s (cid:17) E (cid:16) k sdm P s (cid:17)(cid:16) − P s k rdm P m k sdm (cid:17) (6)and R − m = log( e ) (cid:20) exp (cid:18) k sdm P s (cid:19) E (cid:18) k sdm P s (cid:19)(cid:21) . (7) Since the transmissions from M sources are non-interfering at the destination, an M -1-1 system can beconsidered as independent 1-1-1 systems. The achievable rate for the whole M -1-1 system can then bewritten as, R = M (cid:88) m =1 (cid:0) min (cid:8) R m , R +2 m + R − m (cid:9)(cid:1) . (8)Note that an important assumption in deriving (3) (see [17]), is that the S-R rate is always greater than thesum rates of S-D and R-D (i.e., the relay is able to decode the source signal). When the relay is not ableto decode the source signal, the model assumption is that (e.g., [11][12][17]) either the source is far fromboth the relay and destination or the source is closer to destination than the relay. In both scenarios, zeropower is allocated (i.e., P m = 0 for the m th source) by the relay (using our proposed power allocationscheme). This acts as an admission control mechanism, where only the sources with higher SNR betweenthemselves and the relay are admitted into the system (or allocated power at the relay). Such a schemeis efficient in avoiding wastage of power at the relay, by only admitting sources into the system whoseS-R channel SNR is good (so that the relay is able to decode).Therefore the rate R m > R +2 m + R − m ∀ m . The achievable rate for an M -1-1 DF system with Rayleighfading is then given by, R = min (cid:40) M (cid:88) m =1 R m , M (cid:88) m =1 R +2 m + M (cid:88) m =1 R − m (cid:41) , (9)which can be simplified as R = (cid:80) Mm =1 R m = (cid:80) Mm =1 R +2 m + (cid:80) Mm =1 R − m . Note that when an optimalpower allocation scheme is found at the relay (i.e., the optimal vector P , ..., P m ) the achievable rate in(9) is maximized. In the following section we will develop low-complexity power allocation schemes atthe relay which delivers a system throughput close to the maximum R (denoted as R P AS − ).Note that the channel gains between S-R can be measured by the relay and therefore instantaneousCSI for S-R channels can be obtained at the relay. However, the instantaneous CSI for the channelsbetween the S-D along and the channel between the R-D is not known at the relay. Assuming availabilityof instantaneous CSI of all channels via feedback with zero-delay is no that practical. Therefore, in oursystem model, we have assumed availability of only statistical CSI for all channels, which is a morerealistic setting. However, considering availability of instantaneous CSI for S-R links and availability ofstatistical CSI between S-D and R-D links forms a different hybrid system model. It is important to note that, our proposed power allocation scheme (in the following section) can be easily extended to the suchan hybrid system model. This is apparent from (3), where we need to integrate only the sum rate R m over all channel states (statistical CSI) and all other following results obtained are still applicable.IV. P OWER A LLOCATION S CHEMES AT THE R ELAY
In this section we investigate our power allocation schemes at the relay which obtains a systemachievable rate close to R P AS − (maximum R ). Due to the minimization term in (9), the second term (cid:80) Mm =1 R +2 m + (cid:80) Mm =1 R − m of this equation should be always less than the first term (cid:80) Mm =1 R m , for thepower allocation at the relay to be efficient. Therefore, any power allocation scheme at the relay mustbe under the constraint of (cid:80) Mm =1 R +2 m ≤ (cid:80) Mm =1 R m − (cid:80) Mm =1 R − m . The relay has a total available powerof P r and needs to allocate the power among M users in order to maximize R (i.e., to obtain R P AS − ).Obtaining R P AS − is equivalent to the maximization of R +2 m . The power allocation vector P , ..., P m canbe obtained by solving the following convex optimization problem with the constraints listed below.max { R +2 m } = max P ,...,P m M (cid:88) m =1 log( e ) exp (cid:16) k rdm P m (cid:17) E (cid:16) k rdm P m (cid:17) − exp (cid:16) k sdm P s (cid:17) E (cid:16) k sdm P s (cid:17)(cid:16) − P s k rdm P m k sdm (cid:17) (10)subject to,1) (cid:80) Mm =1 P m = P r ,2) P m ≥ , m = 1 , · · · , M ,3) R +2 m − R m + R − m ≤ , m = 1 , · · · , M .Since our objective is to maximize the throughput in the network, we need to allocate all the poweravailable at the relay and the first constraint (10) is required. Note that in our system model the relayallocates lower power to sources which are closer the destination than the relay. This is because thethroughput increase with the help of a relay will not be significant for sources close to the destination.Note also that the instantaneous CSI is not required for the power allocation at the relay to maximize theachievable rate R . The optimization problem in (10) can be solved using numerical optimization tools.However, in practical relay systems, numerical search algorithm may not be practical. Even the mostefficient optimization search algorithms are known to have complexity of the order O ( M ) (e.g. interiorpoint method [30]). The complexity of such algorithms scales with the number of users, making them intractable. We therefore develop two low-complexity power allocation schemes (PAS-1 and PAS-2) whichcan be easily implemented in a real-time system.Due to the non-linear product exp (cid:16) k rdm P m (cid:17) E (cid:16) k rdm P m (cid:17) in (10), the classical water-filling (CWF) algorithm(e.g. [26]) cannot be used to obtain the power allocation vector. Further, the rate constraint in (10) alsohas a non-linear product involved and an expression for the power limited by the constraint cannot bedirectly obtained. Note that direct application of the CWF algorithm with the mean value of the channelfading coefficients will prove to be sub-optimal (as we will see in section VIII). Our key contributionin this paper is that we develop a rational approximation to the non-linear term exp (cid:16) k rdm P m (cid:17) E (cid:16) k rdm P m (cid:17) , sothat we can solve the optimization problem in (10). Specifically we approximate the non-linear product exp (cid:16) k rdm P m (cid:17) E (cid:16) k rdm P m (cid:17) in (10) by using a rational function of the form, exp (cid:18) k rdm P m (cid:19) E (cid:18) k rdm P m (cid:19) = a m (cid:16) k rdm P m (cid:17) + b m c m + (cid:16) k rdm P m (cid:17) + (cid:15), (11)where a m , b m , c m are constants, and (cid:15) is the error in approximation. The approximation in (11) is basedon minimizing the error (cid:15) . As a measure of the approximation of the estimation of a m , b m and c m , wecomputed the root mean squared error (RMSE) on the approximation as, RMSE = (cid:113) [ S ( a m ,b m ,c m )] n , where S ( a m , b m , c m ) and n are defined in Appendix A. For the approximation in (11), the RMSE was found to be < − (when the SNR at the destination in the range of − to dB) leading to error in approximation (cid:15) < − . The difference between the achievables rates with the approximation and with the exponentialintegral function is therefore < − . Note that in (11), the constants a m , b m and c m depend on the ratio k rdm P m (denoted by ∆ = k rdm P m ). Note also, that for different values of ∆ we may need to find different valuesof the constants a m , b m , c m , which minimizes (cid:15) . We start by evaluating an estimate of the power (denotedas P estm ) allocated to the m th user. P estm is found by setting h srm , h sdm and h rdm to their mean value in (3),and by using a CWF algorithm. Note that P estm is only used to obtain the constants a m , b m , c m so thatthe approximation in (11) can be used. The constants a m , b m and c m , for different ranges of ∆ are thenfound by using the lookup Table I. Appendix A, describes the procedure to obtain Table I. The algorithmfor determining a m , b m and c m is described below. Note that Table. I is pre-computed and stored in the In the above approximation, we have limited the degree of the rational function to 1, as any higher degree rational function leads to apolynomial equation of degree five or higher when we try to solve the Lagrange’s function (discussed later). This leads to an intractablesolution for the power allocation scheme at the relay. Algorithm 1
Algorithm for determining a m , b m and c m . Step 1:
Set h srm = h sdm = h rdm = π √ for all m ∈ { , ..., M } . Note that ∆ , ∆ , ∆ and Ω m are temporaryvariables used in the algorithm. Step 2:
Obtain P estm using the CWF algorithm, under the power constraint (cid:80) Mm =1 P estm = P r . Step 3:
Compute Ω m = 10 log (cid:16) k rdm P estm (cid:17) , for all m ∈ { , ..., M } . For all m ∈ { , ..., M } do the following.If Ω m ∈ ∆ , set a m = a (∆ ) , b m = b (∆ ) and c m = c (∆ ) . Else, if Ω m ∈ ∆ , set a m = a (∆ ) , b m = b (∆ ) and c m = c (∆ ) . Otherwise set a m = a (∆ ) , b m = b (∆ ) and c m = c (∆ ) .memory of the relay.V. L AGRANGIAN -B ASED P OWER A LLOCATION S CHEME (PAS-1)We now develop a Lagrangian-based power allocation scheme at the relay using the approximationgiven in (11). To find the power allocation scheme which maximizes the achievable rate R +2 m in (10), weset up the generalized Lagrange’s multiplier function for non-linear optimization as follows, L ( P, µ, ν, τ ) = − M (cid:88) m =1 log( e ) exp (cid:16) k rdm P m (cid:17) E (cid:16) k rdm P m (cid:17) − exp (cid:16) k sdm P s (cid:17) E (cid:16) k sdm P s (cid:17)(cid:16) − P s k rdm P m k sdm (cid:17) + M (cid:88) m =1 µ m ( − P m ) − M (cid:88) m =1 ν m (cid:0) R +2 m − R m + R − m (cid:1) + (cid:34)(cid:32) M (cid:88) m =1 τ P m (cid:33) − P r (cid:35) , (12)with definitions P = [ P , ..., P M ] , µ = [ µ , ..., µ M ] and ν = [ ν , ..., ν M ] , where µ , ν and τ are Lagrangemultipliers associated with the constraints in (10). We obtain the necessary and sufficient Karush-Kuhn-Tucker (KKT) conditions as, { ∂L ( P,µ,ν,τ ) ∂P m , µ m P m } = 0 {− µ m , − ν m } ≤ ν m (cid:0) R +2 m − R m + R − m (cid:1) ≤ m = 1 , ..., M. (13)To solve the Lagrangian function in (12), we use the approximation in (11). Let φ m ( d rdm , d sdm , P s ) = { [0 , P r ] : φ m ( · ) ∈ R } be a function defined in Appendix B, which denotes the power allocated to the m th user after finding the optimal Lagrange’s multiplier τ ∗ , that satisfies the constraint (cid:80) Mm =1 P m = P r .Similarly, let π cm ( d rdm , d srm , d sdm , P s ) = { [0 , P r ] : π cm ( · ) ∈ R } be another function defined in Appendix B,which denotes the power obtained by using substituting the approximation in (11) into the rate constraintin (10), and solving for P m . We propose the following theorem. Theorem 1 : The power allocation scheme at the relay that approximately obtains the maximum through-put of an M -1-1 system with statistical-CSI is given by, P m = φ m ( d rdm , d sdm , P s ) , if φ m ( d rdm , d sdm , P s ) < π cm ( d rdm , d srm , d sdm , P s ) π cm ( d rdm , d srm , d sdm , P s ) , if φ m ( d rdm , d sdm , P s ) > π cm ( d rdm , d srm , d sdm , P s ) . (14) Proof:
See Appendix B.The PAS-1 algorithm is described below. Even though PAS-1 provides a system achievable rate approx-
Algorithm 2
PAS-1 Algorithm.
Step 1:
Initialize P rem = P r and M ∗ = { , ..., M } . Note that P rem , M ∗ , P ext are variables which arefunction of the iterations between the steps. Step 2:
Obtain a m , b m and c m using Alg. 1 for m ∈ M ∗ . Compute π cm ( · ) . Step 3:
Use bisection search method to compute P m = φ m ( · ) for m ∈ M ∗ subject to (cid:80) m ∈ M ∗ P m = P rem . Step 4:
Find the set M of users, which have φ m ( · ) > π cm ( · ) . If the number of elements in M is equalto either or M , then exit. Step 5:
Let P m = π cm ( · ) , for m ∈ M . Calculate the extra power P ext = (cid:80) m ∈M (cid:2) φ m ( d rdm , d sdm , P s ) − π cm ( · ) (cid:3) . Step 6:
Obtain the set of channels ¯ M complementary to M . For the set ¯ M compute the total power P ¯ M as P ¯ M = (cid:80) m ∈ ¯ M P m . Compute P rem = P ext + P ¯ M and set M ∗ = ¯ M . Goto Step 2.imately equal to R P AS − , as we will see in section VIII, the computations required in the PAS-1 algorithmmay still be high due to the iterative bisection search in Step 2. The system achievable rate is insensitiveto exact power allocation for high values of received SNR at the destination since the achievable rate is alogarithmic function of the power. This motivates us to investigate a lower computational power allocationscheme that can perform close to PAS-1 in the following section.VI. L OW -C OMPUTATIONAL P OWER A LLOCATION S CHEME (PAS-2)Using the results from PAS-1, we now develop another power allocation scheme (PAS-2) at therelay which achieves a system achievable rate within − of the achievable rate R P AS − , but withsignificantly lower computations. To develop PAS-2, we build on the work proposed in [28], in which alow-complexity power allocation scheme was proposed for a single transmitter and receiver system withmulticarrier modulation and Intersymbol Interference (ISI) channels. In [28], the transmitter allocates zero power to subchannels with channel gains greater than a threshold, and equal power to the remainingsubchannels. This concept is motivated by the fact that R is insensitive to exact power allocation forhigh values of SNR. However, the system model considered in [28] is different compared to our system,and as such the power allocation scheme cannot be directly applied to our M -1-1 system. Any powerallocation scheme in our M -1-1 system needs to take into account the rate constraint in (10). Note that thecombined channel gain between the m th source and d ; and the channel between r and d can be obtainedby rearranging the second term of (3) and is given by, G m = ( d sdm ) α (cid:12)(cid:12) h rdm (cid:12)(cid:12) ( d rdm ) α (cid:104) P s | h sdm | + N d ( d sdm ) α (cid:105) . (15)Our power allocation scheme 2 (PAS-2) can be outlined as follows. We compute G m in (15) using themean value of h srm , h sdm and h rdm . We then sort the users based on their channel gains ( G m ) and find the setof users who should be allocated non-zero power. We divide the power equally among the set of users,who must receive non-zero power. We then find the set of users M , whose allocated power exceeds thepower limited by the rate constraint function π cm ( · ) . The remaining power after applying the constraints iscomputed and redistributed equally among the users in the complementary set ¯ M . This is done iterativelyuntil all available power is distributed. The PAS-2 algorithm is described below.VII. C OMPUTATION C OMPLEXITY
A. Computation Complexity of PAS-0 (Optimal)
The exact computational complexity of any numerical method of optimization is difficult to obtain asit depends on the number of times that the objective function and its derivatives are computed. It alsodepends on how many iterations are required to reach some stopping/convergence criterion and how manyconstraints are active during an iteration. In general this will be of the order O ( M ) (e.g. interior pointmethod [30]). B. Computation Complexity of PAS-1
The complexity of the PAS-1 algorithm is mainly in Step 2 and Step 3. Obtaining a m , b m and c m inStep 2 involves the CWF algorithm, whose complexity is of order O ( M log M ) [27]. In Step 2 of PAS-1algorithm, the optimum value of Lagrangian multiplier τ ∗ should be searched to compute φ m ( d rdm , d sdm , P s ) , Algorithm 3
PAS-2 Algorithm.
Step 1:
Initialize P m = 0 , P rem = P r and M ∗ = { , ..., M } . Note that P rem , M ∗ , P ext and P est arevariables which are function of the iterations between the steps. Step 2:
Compute G m for m ∈ { , ..., M } , using (15) with h srm , h sdm and h rdm set to their mean value. Sortthe channel gains, such that G ≥ G ≥ ... ≥ G m for m ∈ M ∗ . Step 3:
Set P m = P rem / | M ∗ | , where | M ∗ | , denotes the cardinality of set M ∗ . Step 4:
If ( /G | M ∗ | ≥ P r + 1 /G ), then | M ∗ | = | M ∗ | − , Goto Step 3. Step 5:
Set P est = P m . Obtain a m , b m and c m from Step 3 of Alg. 1. Compute π cm ( · ) . Step 6:
Set M ∗ = M ∗ + 1 . Find the set M of the channels, which have P m > π cm ( · ) for m ∈ M . If thenumber of elements in M is equal to either or M , then exit. Step 7:
Let P m = π cm ( · ) , for m ∈ M and calculate the extra power as P ext = (cid:80) m ∈M [ P m − π cm ( · )] . Step 8:
Obtain the set of users ¯ M complementary to M . Compute the total power in set ¯ M as P ¯ M = (cid:80) m ∈ ¯ M P m . Compute the remaining power P rem = P ext + P ¯ M and set M ∗ = ¯ M . Goto Step 2.such that the constraint (cid:80) Mm =1 φ m ( d rdm , d sdm , P s ) = P r is satisfied. We used a bisection search method [31]to find τ ∗ . The complexity of an efficient bisection search algorithm is of the order O ( M log M ) [32]. Thetotal complexity of the PAS-1 algorithm is then of the order O ( KM + 2 KM log M ) , where K ( K < M )is the number of iterations required between Step 2 and 7 in the PAS-1 algorithm. Through our simulations(discussed in section VIII), we found that the maximum value of K is K = 20 for M = 100 users. C. Computation Complexity of PAS-2
The PAS-2 algorithm does not require the bisection search of τ ∗ as in the PAS-1 algorithm. Thecomplexity of the sorting of users in Step 1 is O ( M log M ) . The other complexity is in step 3 and 4being iteratively executed. Step 3 and 4 has a complexity of O ( M log M ) [28]. The total computationalcomplexity of the algorithm is then of the order O ( KM + 2 KM log M ) , where K ( K < M ) is thenumber of iterations required between Step 2 and 7 in the PAS-2 algorithm. Here, we will show thatthe PAS-2 algorithm requires far less computations then the PAS-1 algorithm. This is because in Step3 of the PAS-1 algorithm, we need to compute the rate R +2 m for each step of bisection search to findthe new value of τ (see [32]). However, in the PAS-2 algorithm the optimal water-level is found without actually computing R +2 m in each step, and is therefore free of logarithmic operations. Further, during eachiteration, the PAS-2 algorithm is free of logarithm and cube root operations, and requires only 1 squareroot computation per user compared to 8 square roots computations per users in the in Step 2 of thePAS-1 algorithm. The number of computations required per iteration for the PAS-1 and PAS-2 algorithmsare summarized in Table. II. We can see in Table. II that the PAS-2 algorithm requires significantlylower number of computations compared to the PAS-1 algorithm. We also measured the execution timesrequired for each of the power allocation algorithms in MATLAB. The results plotted in Fig. 8 show thatPAS-0 takes over 1000x more time than PAS-1 and PAS-2 algorithms making them suitable for practicalimplementations. VIII. N UMERICAL R ESULTS
Here we present the analytical and simulation results for the system achievable rate using the powerallocation schemes developed in the previous sections. The various parameters were configured as follows.The path-loss exponent was set to α = 2 , P s was set to 5 and N d and N r was set to 1. Note that, wehave investigated the achievable rates for various values of the above parameters. For convenience, wemake the following notations. We denote the system achievable rate obtained with PAS-0 (numericaloptimization), PAS-1 and PAS-2 as R P AS − , R P AS − and R P AS − respectively. We also implemented apower allocation scheme as described in [28] with channel fading coefficients set to their mean value.This corresponding system achievable rate will be denoted as R SUBOP . Note that for PAS-0, we used theInterior-Point Algorithm [30] to obtain R P AS − .We also performed Monte Carlo simulations to verify our analysis. In our simulation setup, thenormalized distance between relay and destination is initially set to 1 and M sources are randomlydistributed, within a circle of radius 0.5 centered around the relay. The relay is then moved along thestraight line towards the destination. For each position of the relay, the system achievable rate is computedas follows. The channel coefficients h srm , h sdm and h rdm are drawn from a Rayleigh distribution. For eachchannel realization, R m for m = 1 to M users is computed from (3) using the PAS-1 algorithm. R wasobtained using (9) and averaged over 2000 channel realizations. The users were redistributed and thesimulations repeated over 2000 trails. The results are plotted in Fig. 3 for M = 5 users and P r = 20 . R P AS − , R P AS − , R P AS − and R SUBOP for M = 25 , P s = 3 and P r = 75 are plotted in Fig. 4. R P AS − , R P AS − , R P AS − for different values of P s is plotted in Fig. 5, with M = 50 and P r = 200 . Note that wehave assumed a subchannel bandwidth of 1MHz (Mega Hertz) and that the achievable rate is in Mbits/sec.Note also that with increasing values of M , R SUBOP is significantly lower than R P AS − , whereas R P AS − is almost equal to R P AS − .The system achievable rate as function of M is plotted in Fig. 6 with relay placed midway on the linebetween the center of the source circle and the destination. Note that with PAS-1, the system achievablerate R P AS − in all the results is approximately equal to R P AS − . R P AS − obtained with PAS-2 is within ≈ of R P AS − . Note that the rates plotted in all the figures is the sum of the achievable rates for M users. The transmit powers at the source and relay and normalized with a reference power of 1 mW.Similiarly the noise powers at the relay and destination are also normalized with a reference power of1mW. We also plotted the results for various values of M , with the relay placed midway between thesource circle and destination, in Fig. 6. Note in Fig. 6, R P AS − is still approximately equal to R P AS − ,whereas R P AS − is still within − of R P AS − . Also note that R SUBOP proves to be sub-optimal.The loss in system throughput with this sub-optimal power allocation is up to ≈ relative to usingPAS-1, for M = 50 users in Fig. 6. We anticipate this loss to be higher for higher values of M . Usingthe PAS-1 or the PAS-2 algorithm, then turns out to be a tradeoff between low computations and systemachievable rate. The system achievable rate with different power allocation schemes for P s = 5 , M = 50 users for different values of P r is shown in Fig. 7. We can see from Fig. 7 that the PAS-1 algorithmdelivers a throughput close to R P AS − and that the PAS-2 algorithm is within − of R P AS − .Therefore, the PAS-1 algorithm is suitable in an M -1-1 system, where the priority is to maximize systemthroughput, whereas the PAS-2 algorithm is suitable for an M -1-1 system where lower computations inpower allocation scheme are essential. IX. C ONCLUSIONS
In this paper, we considered a multi source decode-and-forward, full-duplex relay system (M-1-1 system)with statistical-CSI. We investigated the achievable rate, R , of an M -1-1 system with statistical-CSI,where all channels undergo independent Rayleigh fading. We showed how R can only be maximized using numerical power allocation schemes which has a high-complexity of order O ( M ) . We introduceda rational approximation in the achievable rate analysis, based on which we developed two low-complexitypower allocation schemes at the relay that obtain a system throughput close to the maximum ( R P AS − ).Specifically, we developed a Lagrangian-based power allocation scheme (PAS-1), which obtains a systemachievable rate approximately equal to R P AS − for all practical purposes. Utilizing the results derived inPAS-1, we developed another power allocation scheme (PAS-2), which delivers a system achievable ratewithin − of R P AS − , but with a significantly lower number of computations.Most importantly we showed that the complexity of the PAS-1 and PAS-2 algorithms is of order O ( M log M ) . We provided simulations results to justify our analysis. We showed how PAS-1 is suitablein an M -1-1 system with priority on system throughput, whereas PAS-2 is suitable for an M -1-1 systemwhere lower computations in power allocation scheme are essential. The power allocation schemes, PAS-1 and PAS-2, developed in this paper will be of value to design and implementation of an real-timemulti-user relay systems operating under realistic channel conditions.A CKNOWLEDGMENTS
The authors would like to thank the anonymous reviewers whose input and suggestions greatly improvedthe quality of this paper. The authors would also like to thank the University of New South Wales andthe Australian Research Council (ARC Grant number DP0879401) for supporting this research work.A
PPENDIX AG ENERATING T ABLE
IWe approximated the non-linear product exp (cid:16) k rdm P m (cid:17) E (cid:16) k rdm P m (cid:17) in (10) by a curve fitting technique. Theaverage received SNR (in dB) at the receiver (destination) is given as, γ SNR = P t − P L + G t + G r − N d , where P t = 10 log ( P m ) , P L = α log ( d rdm ) (in dB), and G t and G r are transmit and receive antenna gains (indB), respectively. In the ratio k rdm P m = ( d rdm ) α N d P m , ( d rdm ) α represents the path-loss between the r and d . Therefore,the ratio k rdm P m scales as /γ SNR . Since γ SNR is typically in the range of -15 to 30dB, we are interested infinding the constants a m , b m and c m , when k rdm P m is in the range of − to dB. However, it is not possibleto find one set of values for a m , b m and c m with an acceptable error in approximation (i.e., (cid:15) ≤ − ) overthe entire range of k rdm P m . We therefore divide k rdm P m into 3 ranges as { ∆ ∈ k rdm P m : − dB < ∆ < dB } , { ∆ ∈ k rdm P m : 0 dB < ∆ < dB } and { ∆ ∈ k rdm P m : 15 dB < ∆ < dB } . Note that splitting k rdm P m into more than 3 ranges will although increase the precision of approximation in (11), but would notalter the final results significantly. This is because the error in approximation (cid:15) over all three ranges isalready less than − and increasing the number of ranges k, will although reduce (cid:15) , but will not affect R significantly. The values of a m , b m and c m are found as follows. Let us define, ∆ ik as a discrete value inthe range ∆ k . We perform a non-linear least squares analysis by minimizing the sum of non-linear leastsquares defined as, S ( a, b, c ) = n (cid:88) i =1 (cid:20) exp (cid:0) ∆ ik (cid:1) E (cid:0) ∆ ik (cid:1) − (cid:18) a ∆ ik + bc + ∆ ik (cid:19)(cid:21) , (16)where n is the number of discrete values in ∆ k = [∆ k , ∆ k , ..., ∆ n − k , ∆ nk ] . We used the Levenberg-Marquardt algorithm to minimize the sum of non-linear squares S ( a, b, c ) in 16, by setting n = 10 .Table. I lists the values of a, b , c for different ranges of ∆ k with (cid:15) ≤ × − . We stress here the factthat the values of a, b and c are pre-computed and stored in the memory of the relay. Note that if h sr , h sd and h rd are Rician distributed with parameter ν , then | h sr | , | h sd | , and | h rd | follow non-central χ distribution with two degrees of freedom and non-centrality parameter ν . To find the average achievablerate of the whole system, the channel rates have to be integrated over all channel states, which does notyield a closed form expression. Therefore, the proposed approximation cannot be extended to a channelwith Rician distribution. A PPENDIX BP ROOF FOR P OWER A LLOCATION T HEOREM P m in (13), leads to µ m = ( ν m + τ ) − (log e ) k rdm k sdm (cid:16)(cid:16) a m k rdm + b m P m c m P m + k rdm (cid:17) − β (cid:17) ( P m − k rdm k sdm ) + log e (cid:16) P m − k rdm (cid:16) a m k rdm + b m P m c m P m + k rdm (cid:17)(cid:17) P m ( P m − k rdm k sdm ) , (17) where β = exp (cid:16) k sdm P s (cid:17) E (cid:16) k sdm P s (cid:17) . The condition µ m P m = 0 leads to either P m = 0 or µ m = 0 . Setting µ m = 0 in (17) and after some algebra we get, P m ( τ + ν m ) c m + P m [ k rdm ( τ + ν m ) − c m k rdm k sdm ( τ + ν m ) − c m log e ]+ P m (cid:2) c m ( k rdm k sdm ) ( τ + ν m ) − k rdm ) k sdm ( τ + ν m )+ c m k rdm k sdm (1 − β ) log e + b m k rdm k sdm log e + b m k rdm log e ( b m − (cid:3) + P m [( τ + ν m )( k rdm ) ( k sdm ) + a m ( k rdm ) ( k sdm + 1) log e − k rdm ) k sdm ( b m − β + 1) log e ] − a m ( k rdm ) k sdm = 0 . (18)The rate constraint in (13) leads to two cases, 1) ν m = 0 or 2) (cid:0) R +2 m − R m + R − m (cid:1) = 0 . The secondcase leads to, P m = π cm ( k rdm , k srm , k sdm , P s ) = k − (cid:112) k + 4 k k k . (19)where k = b m − c m ψ , k = k rdm ( a m − ψ ) + c m P s k rdm k sdm ( β − ψ ) , and k = P s ( k rdm ) k sdm ( ψ − β ) , and where, ψ = exp (cid:16) k srm P s (cid:17) E (cid:16) k srm P s (cid:17) , and β = exp (cid:16) k sdm P s (cid:17) E (cid:16) k sdm P s (cid:17) . Setting ν m = 0 in (17) and adding with (18) leadsto ν m + µ m = 0 . But neither ν m or µ m can be lesser than 0 due to the conditions ν m ≥ and µ m ≥ .Thus the multipliers ν m = µ m = 0 . Equation (18) with ν m = 0 is a quartic equation. To solve for P m , weneed to find the roots by first converting the regular quartic into a depressed quartic function of the form, P m + λ P m + λ P m + λ P m − λ = 0 , (20)where, λ = − k rdm (2 c m k sdm − c m − τ log e , (21) λ = k rdm c m (cid:20) k sdm ( c m k rdm k sdm −
1) + k sdm [ c m (1 − β ) + b m ] τ log e + k rdm [ b m − τ log e (cid:21) , (22) λ = k rdm k sdm (cid:2) ( k rdm ) k sdm τ log e + b m + (1 − β ) (cid:3) c m τ log e + ( k rdm ) a m c m τ log e , (23)and λ = ( k rdm ) k sdm a m c m τ log e . (24) P m is then, one of the roots to the quartic function [29] in (20). The four roots of the quartic functionare given by, φ m ( d rdm , d sdm , P s ) = η ± η − λ − η ± η − λ (25) where, η = (cid:114) λ − λ + θ, η = (cid:114) λ − η − λ + ( λ λ − λ − λ ) η if η (cid:54) = 0 (cid:113) λ − λ + √ θ − λ if η = 0 , (26)and η = (cid:114) λ − η − λ − ( λ λ − λ − λ ) η if η (cid:54) = 0 (cid:113) λ − λ − √ θ − λ if η = 0 . (27)In the above equations, θ is defined as θ = λ − √ − λ + 3 λ λ − λ ) (cid:113) υ + (cid:112) υ + υ + (cid:113) υ + (cid:112) υ + υ √ , (28)where, υ = 2 λ − λ λ λ + 27 λ + 27 λ λ − λ λ , and υ = 4(3 λ λ − λ − λ ) . Note that thereare four possible roots for φ m ( d rdm , d sdm , P s ) in (25). A closer inspection reveals that not all the roots areuseful. This is because in (20), the coefficient λ defined in (24) is always greater than 0, since k rdm , k sdm and τ are greater than 0. From Descrates’s sign rule [29], there are either three roots or only one positiveroot for the quartic function in (20), irrespective of the sign of λ , λ or λ . The first root in (25), ispositive, since, η , η and η are ≥ and λ ≤ . When three positive roots are available, the second,third and fourth root in (25) are closer to or less than zero when the multiplier τ < . Since a small valueof τ is desirable [28], only the first root in (25) is useful. P m is then given by, P m = φ m ( d rdm , d sdm , P s ) = η + η − λ . (29)The power allocation vector P m can be then be summarized as in (14). The optimal Lagrange’s multiplier τ ∗ , which maximizes (10), can be found through a one-dimensional search (e.g. using bisection [31]),such that the constraint (cid:80) Mm =1 φ m ( d rdm , d sdm , P s ) = P r is satisfied. Note that when all the sources are atequal distances to the relay and destination, k sdm , k srm and k rdm are equal for all sources m = 1 , ..., M . Thisthen leads to the parameters λ , λ , λ , λ , η , η , η , θ , υ and υ to be equal for all sources, for anyvalue of the multiplier τ . Due to the constraint, (cid:80) Mm =1 φ m ( d rdm , d sdm , P s ) = P r , equal power is allocated toall the sources. R EFERENCES [1] The 802.16j-Relay Task Group, “Air interface for fixed and mobile broadband wireless access systems: Multihop relay specification”,IEEE P802.16j/D9, 2009. [2] A. Nosratinia, A. Hedayat, “Cooperative Communication in wireless networks,” IEEE Comm. Mag ., vol. 42, no. 10, pp. 7480, Oct. 2004.[3] A.Chakrapani, et.al, “Low complexity power allocation scheme for regenerative multi-user relay networks,” in Proc. IEEE InternationalConference on Communications (ICC), Sydney, NSW, 2014 , pp. 5419-5425[4] T. Riihonen, et.al, “Comparison of full-duplex and half-duplex modes with a fixed amplify-and-forward relay,” in Proc. IEEE WirelessComm. and Networking Conf. , Apr. 2009.[5] T. Riihonen, et.al, “Comparison of full-duplex and half-duplex modes with a fixed amplify-and-forward relay,” in Proc. IEEE WirelessComm. and Networking Conf. , Apr. 2009.[6] T. Riihonen, et.al, “On the feasibility of full-duplex relaying in the presence of loop interference,” in Proc. 10th IEEE Workshop onSignal Proc. Advances in Wireless Comm. , Jun. 2009.[7] B. Chun, et.al, “Pre-nulling for self-interference suppression in full-duplex relays,” in Proc. APSIPA, Ann. Summit and Conf. , Oct. 2009.[8] M. Duarte, A. Sabharwal. “Full-Duplex Wireless Communications Using Off-The-Shelf Radios: Feasibility and First Results,” th Asilomar Conf. on Signals, Sys., and Comps. , 2010.[9] J. I. Choi, et.al, “Achieving single channel, full duplex wireless communication,” in Proc. of MOBICOM , 2010, pp.1-12.[10] R. Nikjah, N. C. Beaulieu, “Achievable Rates and Fairness in Rate-less Coded Decode-and-Forward Half-Duplex and Full-DuplexOpportunistic Relaying”,
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Relay ( r ) d S S S S M . . . . Achievable rate R Fig. 1. Multi-source S m , m = { , ..., M } , with single relay and destination ( M -1-1) system. Relay r d S m (cid:1) (cid:2)(cid:3)(cid:4) (cid:1) (cid:2)(cid:3)(cid:5) (cid:1) (cid:2)(cid:5)(cid:4) (cid:6) (cid:7) (cid:7) (cid:7) (cid:6) (cid:6) (cid:8)(cid:9)(cid:10), (cid:12) (cid:4) (cid:13) (cid:8)(cid:9)(cid:10), (cid:12) (cid:4) (cid:13) (cid:8)(cid:9)(cid:10), (cid:12) (cid:5) (cid:13) Fig. 2. The m th source S m , relay and destination form a 1-1-1 triangle model. TABLE IR
ATIONAL FUNCTION CONSTANTS FOR DIFFERENT RANGES OF ∆ k , k ∈ { , , }
10 log (∆ k ) in dB a b c ∆ = {− to } dB 2.4989 0.0364 0.005416 ∆ = { to } dB 0.3495 0.3698 0.0985 ∆ = { to } dB 0.003246 0.9306 0.583TABLE IIN UMBER OF C OMPUTATIONS R EQUIRED FOR THE
PAS-1
AND
PAS-2
ALGORITHMS PER ITERATION .Operation PAS-1 Algorithm PAS-2 AlgorithmMult. M log M + 64 M M log M + 10 M Div. M log M + 15 M M log M + 3 M log( · ) M log M exp( · ) 2 M ME ( · ) 2 M M (cid:112) ( · ) 8 M M (cid:112) ( · ) 2 M Distance between Relay and Destination A c h e i v a b l e R a t e i n M b i t s / S ec For
M = P s = P r = R PAS−0 − (Optimal) R PAS−1 − proposed R PAS−2 − proposed R SUBOP R PAS−1 − Simulation
Fig. 3. Achievable Rates with different power allocation schemes for M = 5 users, P s = 5 and P r = 20 . Distance between Relay and Destination A c h i e v a b l e R a t e i n M b it s / S ec For
M = P s = P r = R PAS−0 − (Optimal) R PAS−1 − proposed R PAS−2 − proposed R SUBOP R PAS−1 − Simulation
Fig. 4. Achievable Rates with different power allocation schemes for M = 25 users, P s = 3 and P r = 75 . Power at Source P s A c h i e v ab l e R a t e i n M b i t s / S e c For M = 50 users and P r = 150 R PAS − - (Optimal) R PAS − - Proposed R PAS − - Proposed R SUBOP
Fig. 5. Achievable Rates with different power allocation schemes plotted against P s for M = 50 users and P r = 200 . Number of users M A c h i e v a b l e R a t e i n M b it s / S ec For P r = M*4 R PAS−0 − (Optimal) R PAS−1 − proposed R PAS−2 − proposed R SUBOP P s = 1 P s = 5 Fig. 6. Achievable Rates with different power allocation schemes for different M with P r = 4 M , plotted for P s = 1 and P s = 5 .
100 110 120 130 140 150 160 170 180 190 200150155160165170175180185190195200
Total Power at Relay P r A c h i e v a b l e R a t e i n M b it s / S ec For M = 50 users with P s = 5. R PAS − - (Optimal) R PAS − - Proposed R PAS − - Proposed R SUBOP
Fig. 7. Achievable Rates with different power allocation schemes plotted against P r for M = 50 users and P s = 5 .